Analytical approximate periodic solutions for two-degree-of-freedom coupled van der Pol-Duffing oscillators by extended homotopy analysis method

Qian, Youhua; Zhang, W.; Lin, Bin; Lai, Shih-Kung

doi:10.1007/s00707-010-0433-3

Cited by 11 publications

(2 citation statements)

References 36 publications

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“…Kargarnovin et al [16] applied homotopy analysis method to analyze the nonlinear free vibrations of the simple end beams. Qian et al [17,18] employed homotopy analysis method to obtain approximate solutions for an electrostatically actuated microbeam and an elastically restrained beam with a lumped mass, respectively. Wu et al [19] contributed to the research of the nonlinear thickness-shear vibrations of a finite crystal plate with homotopy analysis method.…”

Section: Shock and Vibrationmentioning

confidence: 99%

Approximate Series Solutions for Nonlinear Free Vibration of Suspended Cables

Zhao

Sun²,

Wang³

et al. 2014

Shock and Vibration

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This paper presents approximate series solutions for nonlinear free vibration of suspended cables via the Lindstedt-Poincare method and homotopy analysis method, respectively. Firstly, taking into account the geometric nonlinearity of the suspended cable as well as the quasi-static assumption, a mathematical model is presented. Secondly, two analytical methods are introduced to obtain the approximate series solutions in the case of nonlinear free vibration. Moreover, small and large sag-to-span ratios and initial conditions are chosen to study the nonlinear dynamic responses by these two analytical methods. The numerical results indicate that frequency amplitude relationships obtained with different analytical approaches exhibit some quantitative and qualitative differences in the cases of motions, mode shapes, and particular sag-to-span ratios. Finally, a detailed comparison of the differences in the displacement fields and cable axial total tensions is made.

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Section: Shock and Vibrationmentioning

confidence: 99%

Approximate Series Solutions for Nonlinear Free Vibration of Suspended Cables

Zhao

Sun²,

Wang³

et al. 2014

Shock and Vibration

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“…Qian et al [34] extended the model by assuming a linear elastic and a linear damping coupling for two oscillators: van der Pol and Duffing ones. The extended homotopy analysis method is applied to derive the accurate approximate analytical solutions for the two degree of freedom coupled van der Pol -Duffing oscillator:  1 and  2 are bifurcation parameters,  is the detuning parameter that is proportional to the difference of the oscillator frequencies,  is the non-isochronisms parameter that relies on the oscillation frequency and amplitude,  and  are the coupling inertial and dissipative parameters.…”

Section: Coupled Oscillatorsmentioning

confidence: 99%

On the Van Der Pol Oscillator: an Overview

Cvetićanin

2013

AMM

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In this paper an overview of the self-sustained oscillators is given. The standard van der Pol and the Rayleigh oscillators are considered as basic ones. The cubic nonlinear term of Duffing type is included. The special attention is given to the various complex systems based on the Rayleighs and van der Pols oscillator which are extended with the nonlinear oscillators of Duffing type and also excited with a periodical force. The connection is with the linear elastic force or with linear damping force. The objectives for future investigation are given in this matter.

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